Pretty Pictures

[Damian]

How about adding color to the plot?
It could be something like start with blue, and become red as it reaches the end of the loop.
And a blue circle in the first position and a red circle in the end one.
That way one can se the "path" made by the ant.

[Orgasmic Troll]

Originally Posted by Damian

How about adding color to the plot?
It could be something like start with blue, and become red as it reaches the end of the loop.
And a blue circle in the first position and a red circle in the end one.
That way one can se the "path" made by the ant.

Any takers? This seems like it might be doable, but I'm not sure how to do it

[potonono]

Attached is a 3D color spread for the first 50000 primes. Starts at bright red, fades into purple, violet, and eventually blue. There is a black cylinder from the start point (red) to the end point (blue).

This was made in POV-Ray 3.6 (freeware)

[Damian]

Looks pretty!
Just one question, as the distance between two primes gets longer as x increases, shouldn't the straight lines also get longer as the ant advances in it's path?
In that case, and looking in the diagram, it looks like the ant started in the blue region and ended in the red one.
(because there are longer straight lines in the red section)

Originally Posted by potonono

Attached is a 3D color spread for the first 50000 primes. Starts at bright red, fades into purple, violet, and eventually blue. There is a black cylinder from the start point (red) to the end point (blue).

This was made in POV-Ray 3.6 (freeware)

[potonono]

Woops! You are correct. I mislabelled my colors. You would start in the blue and move toward red. Attached are a couple zooms on the original and final position.

[potonono]

final position

[Damian]

At first I thought that it was a 2D plot you made.
Did you use Uncwilly's sugestion for making it 3D? I would also like to see the 2D version.

Also I have a new question now
(both in the 2D and the 3D version)
Lets call Path(t) the path the ant makes from 1 to t.
I wonder for what value of t the ant first returns to the origin (if it ever does).
Or more generally what is the number of times the ant returns to some point (x,y) or (x,y,z)

And a 3D variant: it works equal to the 2D version but when the ant changes direction (at a prime) it would also jump up in the +z direction.
That way the +z jumps will be counting the number of primes.

And how about a 2D graph R(t) of the distance of the ant at Path(t) to the origin?

Now everyone has homework!

[geoff]

Originally Posted by Damian

I wonder for what value of t the ant first returns to the origin (if it ever does).

It never does. The ant only ever takes one step of odd length (primes 2 and 3 are the only primes an odd distance apart), but to return to the origin it must take an even number of odd steps.

Or more generally what is the number of times the ant returns to some point (x,y) or (x,y,z)

I'll leave the hard questions for someone else :-)

[Damian]

What can be seen now is that the ant never goes over a position where both x and y have pair values. (And so it doesn't return to (0,0)).
And both x and y have to be odd for there been a chance that t is prime (and thus the ant turn to the left)
If only one value (x or y) is odd, and the other is pair, then t is pair for that value of Path(t)

A new question would be: for what value of t the ant first return to the first prime position, if it ever does? (that is Path(2)). If it does then t will be pair because we are talking about (1,0).

Originally Posted by geoff

It never does. The ant only ever takes one step of odd length (primes 2 and 3 are the only primes an odd distance apart), but to return to the origin it must take an even number of odd steps.



I'll leave the hard questions for someone else :-)

[Orgasmic Troll]

Originally Posted by Damian

And how about a 2D graph R(t) of the distance of the ant at Path(t) to the origin?

Now everyone has homework!

Here's the 2d version:

[Damian]

Originally Posted by TravisT

Here's the 2d version:

From looking at your graph the ant didn't return to Path(2) at last for t < 50.000 (it's value should be 1 at that point)

I now installed Mathematica so I can make some graphics too.
I will draw the 1D case (the ant walks over a 1 dimensional line, and turn around at prime values of t )
I draw path(t) as a function of t (thus a 2D graph).

Here's the code:

ptslist2 = {0, 1}; pt2 = 1;
For[i = 1, i < 50000, i = i + 2, pt2 = pt2 - (Prime[i + 1] - Prime[i]);
ptslist2 = Append[ptslist2, pt2];
pt2 = pt2 + (Prime[i + 2] - Prime[i + 1]);
ptslist2 = Append[ptslist2, pt2]];
ListPlot[ptslist2, PlotJoined -> True]